The bisection method
The bisection method is considered the simplest one-dimensional root-finding algorithm. The general interest is to find the value of a continuous function
such that
.
Suppose we know the two points of an interval and
, where
, and that
and
lie along the continuous function, taking the midpoint of this interval as
, where
, the bisection method then evaluates this value as f(c).
Let's illustrate the setup of points along a nonlinear function with the following graph:
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Since the value of f(a) is negative and f(b) is positive, the bisection method assumes that the root lies somewhere between a and b and gives
.
If or is very close to zero by some predetermined error tolerance value, then a root is declared as found. If
, then we may conclude that a root exists along the interval
and
, or interval
and
otherwise.
On the next evaluation, is replaced as either
or
accordingly. With the new interval shortened, the bisection method repeats with the same evaluation to determine...